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title: “HW1.RMD” author: “Mara” date: “3/8/2023” output: html_document

HOMEWORK 1

R Markdown Syntax

Picture

High Kick during a Muay Thai Fight

Quote

“the realm of birth and death”

Emojis

Look at my nice emojis: 💩 🎈 ❤️

Giphy

Here is a big Giphy:

Is that a jojo reference?

and a smaller one…

Tables and Columns

Experiment with the Rmd Code below and test output.

Classes	Time
International Macroeconomics	4 hours
Microéconomie II	2 hours
Eco contemporaine Dans une perspective Historique	4 hours
Création d’entreprise et système d’innovation	2 hours
Intro to Data Science	4 hours
Intro à la fiscalité	2 hours

Video

R Markdown Syntax

Show an example where the chunk option cache = T leads to a misleading answer. This example must be different from the one presented in the textbook.

a = 20

(m = a/2)

## [1] 10

here, if we change the value of a, the output remains the cached initial a divided by 2.

set.seed(100)
samples <- runif(n=100, min=0, max=1)
x <- samples
print(x)

theoreticalmean <- ((0+1)/2)
theoreticalvar <- ((1-0)/12)

mean(x)
median(x)
var(x)

the observed values are slighty off. The mean and median are relatively close to the theoretical ones, which makes sense because we generate a large number of samples between 0 and 1, which correlates with the law of large numbers. The variance is slightly off, which makes sense because we do not controll where the samples are located between 0 and 1, so we might get a larger variance.

hist(x, xlab = "uniformly dist. 100 random samples")

Equations

Here we have centered equations

\[\frac{\partial\ l }{\partial\beta }= \sigma _{}^{-2} \sum_{}^{} \mathbf{X_{i}^{'}} \mathbf{V_{i}^{-1}}(\mathbf{y_{i}-X_{i}\beta })\]

\[\frac{\partial l}{\partial \sigma ^2}=-\frac{1}{2}N_T\sigma ^{-2}+\frac{1}{2}\sigma^{-4}\mathbf{\sum (y_i-X_i\beta)'V_i^{-1}(y_i-X_i\beta)}\]

\[\frac{\partial l}{\partial \mathbf{D}}=-\frac{1}{2} \mathbf{\sum \mathbf{ [Z_\textit{i}^{'}V_\textit{i}^{-1}Z_\textit{i}-\sigma ^{-2}Z_\textit{i}^{'}V_\textit{i}^{-1}(y_\textit{i}-X_\textit{i}\beta)(y_\textit{i}-X_\textit{i}\beta)^{'}V_\textit{i}^{-1}Z_\textit{i}]}}\] And a formula that is embedded in the text: \(\mathbf{A}\equiv [a_{i,j}]_{i,j=1,...,p}\) - what a nice formula!

Quote in Red

“Mieux que l’amour, l’argent, la gloire, donnez moi la vérité.”, Henri David Thoreau

A Button and a Box!

Surprise!

6 is the grade we want ;)

Very Important ! 🎈

EXERCISE 2 - FINANCIAL ANALYSIS

Questions 1 and 2

mydates <- c("01/01/2022","01/01/2023")

mydates2 <- as.Date(mydates, format = "%m/%d/%Y")
mydates2

## [1] "2022-01-01" "2023-01-01"

getSymbols("AAPL", from = min(mydates2), to = max(mydates2))

## [1] "AAPL"

getSymbols("CSCO", from = min(mydates2), to = max(mydates2))

## [1] "CSCO"

getSymbols("INTC", from = min(mydates2), to = max(mydates2))

## [1] "INTC"

getSymbols("HD", from = min(mydates2), to = max(mydates2))

## [1] "HD"

getSymbols("GOOG", from = min(mydates2), to = max(mydates2))

## [1] "GOOG"

getSymbols("JPM", from = min(mydates2), to = max(mydates2))

## [1] "JPM"

Question3:

plot(AAPL$AAPL.Close)

plot(CSCO$CSCO.Close)

Question 4

mydatesnew <- c("10/01/2022","01/01/2023")
mydates3 <- as.Date(mydatesnew, format = "%m/%d/%Y")
mydates3

## [1] "2022-10-01" "2023-01-01"

getSymbols("GOOG", from = min(mydates3), to = max(mydates3))

## [1] "GOOG"

getSymbols("JPM", from = min(mydates3), to = max(mydates3))

## [1] "JPM"

candleChart(GOOG, theme = 'white', type = 'candles')

candleChart(JPM, theme = 'white', type = 'candles')

Question 5

mean(AAPL$AAPL.Close)

## [1] 154.8351

mean(GOOG$GOOG.Close)

## [1] 95.43667

mean(JPM$JPM.Close)

## [1] 126.7063

mean(CSCO$CSCO.Close)

## [1] 48.53781

mean(INTC$INTC.Close)

## [1] 38.5945

mean(HD$HD.Close)

## [1] 310.3346

var(AAPL$AAPL.Close)

##            AAPL.Close
## AAPL.Close   170.4613

var(GOOG$GOOG.Close)

##            GOOG.Close
## GOOG.Close   26.39921

var(JPM$JPM.Close)

##           JPM.Close
## JPM.Close  96.17719

var(CSCO$CSCO.Close)

##            CSCO.Close
## CSCO.Close   31.87959

var(INTC$INTC.Close)

##            INTC.Close
## INTC.Close    80.1732

var(HD$HD.Close)

##          HD.Close
## HD.Close 842.1292

Question 6

mean(AAPL$AAPL.Close)

## [1] 154.8351

mean(GOOG$GOOG.Close)

## [1] 95.43667

mean(JPM$JPM.Close)

## [1] 126.7063

mean(CSCO$CSCO.Close)

## [1] 48.53781

mean(INTC$INTC.Close)

## [1] 38.5945

mean(HD$HD.Close)

## [1] 310.3346

sd(AAPL$AAPL.Close)

## [1] 13.05608

sd(AAPL$AAPL.Close)

## [1] 13.05608

sd(GOOG$GOOG.Close)

## [1] 5.138016

sd(JPM$JPM.Close)

## [1] 9.806997

sd(CSCO$CSCO.Close)

## [1] 5.646201

sd(INTC$INTC.Close)

## [1] 8.953949

sd(HD$HD.Close)

## [1] 29.01946

Question 7

getSymbols("AAPL", from = min(mydates2013), to = max(mydates2013))

## [1] "AAPL"

getSymbols("CSCO", from = min(mydates2013), to = max(mydates2013))

## [1] "CSCO"

getSymbols("INTC", from = min(mydates2013), to = max(mydates2013))

## [1] "INTC"

getSymbols("HD", from = min(mydates2013), to = max(mydates2013))

## [1] "HD"

getSymbols("GOOG", from = min(mydates2013), to = max(mydates2013))

## [1] "GOOG"

getSymbols("JPM", from = min(mydates2013), to = max(mydates2013))

## [1] "JPM"

AAPL_returns <- na.omit(ClCl(AAPL))
GOOG_returns <- na.omit(ClCl(GOOG))
CSCO_returns <- na.omit(ClCl(CSCO))
INTC_returns <- na.omit(ClCl(INTC))
HD_returns <- na.omit(ClCl(HD))
JPM_returns <- na.omit(ClCl(JPM))
Rf <- 0.03/(252)

mean(AAPL_returns) mean(GOOG_returns) mean(CSCO_returns) mean(INTC_returns) mean(HD_returns) mean(JPM_returns)

sd(AAPL_returns) sd(GOOG_returns) sd(CSCO_returns) sd(INTC_returns) sd(HD_returns) sd(JPM_returns)

sharpe_ratioAAPL <- (mean(AAPL_returns) - Rf) / sd(AAPL_returns) * sqrt(252) sharpe_ratioGOOG <- (mean(GOOG_returns) - Rf) / sd(GOOG_returns) * sqrt(252) sharpe_ratioCSCO <- (mean(CSCO_returns) - Rf)/sd(CSCO_returns) * sqrt(252) sharpe_ratioINTC <- (mean(INTC_returns) - Rf )/ sd(INTC_returns) * sqrt(252) sharpe_ratioHD <- (mean(HD_returns)- Rf) / sd(HD_returns) * sqrt(252) sharpe_ratioJPM <- (mean(JPM_returns)- Rf) / sd(JPM_returns) * sqrt(252)

cat(“Sharpe Ratios for each stock:”) print(sharpe_ratioAAPL) print(sharpe_ratioGOOG) print(sharpe_ratioCSCO) print(sharpe_ratioINTC) print(sharpe_ratioHD) print(sharpe_ratioJPM)

AAPL_return = quantmod::dailyReturn(AAPL) CSCO_return = quantmod::dailyReturn(CSCO) INTC_return = quantmod::dailyReturn(INTC) HD_return = quantmod::dailyReturn(HD) GOOG_return = quantmod::dailyReturn(GOOG) JPM_return = quantmod::dailyReturn(JPM)

#excess return AAPL_excess = AAPL_return - 0.03/252 CSCO_excess = CSCO_return - 0.03/252 INTC_excess = INTC_return - 0.03/252 GOOG_excess = GOOG_return - 0.03/252 JPM_excess = JPM_return - 0.03/252 HD_excess = HD_return - 0.03/252

#yearly sharpe ratio AAPL_sharpe_yearly = mean(AAPL_excess) / sd(AAPL_excess) * sqrt(252) CSCO_sharpe_yearly = mean(CSCO_excess) / sd(CSCO_excess) * sqrt(252) INTC_sharpe_yearly = mean(INTC_excess) / sd(INTC_excess) * sqrt(252) HD_sharpe_yearly = mean(HD_excess) / sd(HD_excess) * sqrt(252) GOOG_sharpe_yearly = mean(GOOG_excess) / sd(GOOG_excess) * sqrt(252) JPM_sharpe_yearly = mean(JPM_excess) / sd(JPM_excess) * sqrt(252)

yearly_shape = cbind(AAPL_sharpe_yearly, CSCO_sharpe_yearly, INTC_sharpe_yearly, HD_sharpe_yearly, JPM_sharpe_yearly)

colnames(yearly_shape) = c(“AAPL”, “CSCO”, “INTC”, “HD”, “GOOG”, “JPM”) yearly_shape

#estimation of mean (mu) and variance (Sigma) mat_return = cbind(AAPL_return, CSCO_return, INTC_return, HD_return, GOOG_return, JPM_return) colnames (mat_return) = c(“AAPL”, “CSCO”, “INTC”, “HD”, “GOOG”, “JPM”)

mu = apply(mat_return, 2, mean) Sigma = var(mat_return) ```

equally weighted portfolio

equal_return = (AAPL_return + CSCO_return + INTC_return + HD_return + GOOG_return + JPM_return)/6

muequal = mean(equal_return) varequal = var(equal_return)

#The individual variance of these 6 stocks are: diag(Sigma